Narrow Results

Let Q denote a quadratic polynomial and A_∞ the super-attracting basin of Q at the point ∞ on the Riemann sphere . There exists a unique Riemann mapping Φ from the open disk onto A_∞ such that Φ(∞)=∞, Φ'(∞)=1 and Φ^-1 conjugates Q:A_∞→A_∞ to the squaring map S:D→D:z↦z². In this paper, we show if Q is real and infinitely renormalizable of bounded type then the continuous extension of Φ to the closed disk cannot have any Hölder continuity on the boundary circle .

The goal of this paper is to investigate the iterative behavior of a particular class of rational functions which arise from Newton's method applied to the entire function (z² + c)e^Q(z) where c is a complex parameter and Q is a nonconstant polynomial with deg(Q) ≤ 2. In particular, the basins of attracting fixed points will be described.

In this paper, we consider the family of rational maps given by

In this paper, we first establish a rational iteration method which can be used as a root-finding algorithm for almost every polynomial. It has no nonrepelling extraneous fixed point in the complex plane and is generally convergent for both quadratic and cubic polynomials. Then some properties of this algorithm are given. By the aid of computer, we produce pictures of the Julia sets for the iterations of some polynomials. Numerical results show that it is a root-finding method with convergence order the same as Halley's method.

In this work we present the alternated Julia sets, obtained by alternated iteration of two maps of the quadratic family and prove analytically and computationally that these sets can be connected, disconnected or totally disconnected verifying the known Fatou–Julia theorem in the case of polynomials of degree greater than two. Some examples are presented.